Geometrical insight into non-smooth bifurcations of a soft impact oscillator

Haibo Jiang, Marian Wiercigroch*

*Corresponding author for this work

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

In this paper, the concept of discontinuity geometry of rigid impact oscillators is extended to soft impact oscillators and the mechanisms of grazing bifurcations of an impact oscillator with one-sided elastic constraint are studied by the geometric and dynamical systems methods. The existence conditions of periodic solutions are given by the discontinuity geometry objects and then are used to derive the discontinuity curves. Several bifurcation scenarios near grazing bifurcation are studied to examine the evolution of the system dynamics from a geometrical point of view. The geometrical conditions are obtained for the existence of periodic orbits with one impact, grazing and saddle-node bifurcations. Some geometrical insight is gained into a question whether there is a discontinuous jump or a continuous transition from a non-impacting period-1 to an impacting period-1 attractor at a grazing bifurcation.

Original languageEnglish
Pages (from-to)662-678
Number of pages17
JournalIMA Journal of Applied Mathematics
Volume81
Issue number4
Early online date8 Mar 2016
DOIs
Publication statusPublished - 2016

Fingerprint

Grazing Bifurcation
Dynamical systems
Bifurcation
Discontinuity
Geometry
Orbits
Dynamical Systems Method
Saddle-node Bifurcation
System Dynamics
Periodic Orbits
Attractor
Periodic Solution
Jump
Scenarios
Curve

Keywords

  • Discontinuity geometry
  • Grazing induced bifurcations
  • Impact oscillators
  • Non-smooth systems

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Geometrical insight into non-smooth bifurcations of a soft impact oscillator. / Jiang, Haibo; Wiercigroch, Marian.

In: IMA Journal of Applied Mathematics, Vol. 81, No. 4, 2016, p. 662-678.

Research output: Contribution to journalArticle

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note = "Funding This work is partially supported by the National Natural Science Foundation of China (Grant Nos 11402224, 11202180, 61273106, 11171290), the Qin Lan Project of the Jiangsu Higher Education Institutions of China, and the Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-aged Teachers and Presidents. Acknowledgements H.J. acknowledges the hospitality of the University of Aberdeen. We also thank Prof. Yoshisuke Ueda and Dr Yang Liu for helpful comments.",
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AB - In this paper, the concept of discontinuity geometry of rigid impact oscillators is extended to soft impact oscillators and the mechanisms of grazing bifurcations of an impact oscillator with one-sided elastic constraint are studied by the geometric and dynamical systems methods. The existence conditions of periodic solutions are given by the discontinuity geometry objects and then are used to derive the discontinuity curves. Several bifurcation scenarios near grazing bifurcation are studied to examine the evolution of the system dynamics from a geometrical point of view. The geometrical conditions are obtained for the existence of periodic orbits with one impact, grazing and saddle-node bifurcations. Some geometrical insight is gained into a question whether there is a discontinuous jump or a continuous transition from a non-impacting period-1 to an impacting period-1 attractor at a grazing bifurcation.

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